It is conceivable that a Harnack inequality for Dirichlet forms can be proved for an equation with fixed order, with constants that do not blow up as the order goes to two, using the ideas from the Holder estimates<ref name="K"/>.

It is conceivable that a Harnack inequality for Dirichlet forms can be proved for an equation with fixed order, with constants that do not blow up as the order goes to two, using the ideas from the Holder estimates<ref name="K"/>.

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* An equation where the Harnack inequality fails.

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* '''An equation where the Harnack inequality fails.'''

There exist integro-differential equations of the form

There exist integro-differential equations of the form

Latest revision as of 15:57, 23 April 2015

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The Harnack inequality refers to a control of the maximum of a nonnegative solution of an equation by its minimum. Unlike the local case (either De Giorgi-Nash-Moser theorem or Krylov-Safonov theorem), for nonlocal equations one needs to assume that the function is nonnegative in the full space.

The Harnack inequality is tightly related to Holder estimates for solutions to elliptic/parabolic equations. For a large class of problems both statements are equivalent. But there are simple cases (stable processes with the spectral measure consisting of atoms) where the Harnack inequality fails but Hölder estimates still hold true.

The result can hold either in the parabolic or elliptic setting. The parabolic Harnack inequality trivially implies the elliptic one. The reverse implication is not automatic, and the proof in the parabolic case may have some extra difficulties compared to the elliptic case.

In this case both the elliptic and parabolic Harnack inequality is known to hold with a constant $C$ which does not blow up as $s\to 2$, and $\|f\|$ refers to $\|f\|_{L^\infty(B_1)}$ [1][2]. It is a generalization of Krylov-Safonov theorem.

Elliptic equations with variable order (but strictly less than 2). If $L_x u(x)$ is an integro-differential operator of the form

\[ L_x u(x) = \int_{\R^n} (u(x+y)-u(x)- y \cdot \nabla u(x) \chi_{B_1}(y)) K(x,y) \dd y \]
with uniformly elliptic of variable order: $\lambda |y|^{-n-s_1} \leq K(x,y) \leq \Lambda |y|^{-n-s_2}$ and $0<s_1 < s_2 < 2$ and $s_2 - s_1 < 1$, then
the elliptic Harnack inequality holds if $f \equiv 0$[3]. The constants in this result blow up as $s_2 \to 2$, so it does not generalize Krylov-Safonov theorem. The proof uses probability and was based on a previous result with fixed order [4].

It is conceivable that a purely analytic proof could be done using the method of the corresponding Holder estimate[5], but such proof has never been done.

Gradient flows of symmetric Dirichlet forms with variable order. If $u_t - L_x u(x)=0$ is the gradient flow of a Dirichlet form:

It is conceivable that a Harnack inequality for Dirichlet forms can be proved for an equation with fixed order, with constants that do not blow up as the order goes to two, using the ideas from the Holder estimates[7].

An equation where the Harnack inequality fails.

There exist integro-differential equations of the form
\[ \int_{\R^n} (u(x+y)-u(x)) K(y) \dd y = 0,\]
for which the Harnack inequality fails. In this case the kernels $K$ can be nonnegative, stable (i.e $K(ry) = r^{-n-s} K(y)$), symmetric (i.e. $K(y) = K(-y)$), independent of $x$, bounded above by
\[ K(y) \leq \Lambda |y|^{-n-s},\]
and satisfying the bound below only in a cone of directions
\[ K(y) \geq \lambda |y|^{-n-s} \mathbb{1}_S(y),\]
where $S$ is a cone with positive measure on $\partial B_1$.

The construction was first given in [8]. See also the discussion in [9]. The example is particularly relevant because Holder estimates hold for this equation.